Should I give the insurance company a recorded statement?

Use caution. The at-fault party’s insurer may ask questions designed to test liability, causation, prior conditions, or the seriousness of the injury, and a recorded answer can later be compared with medical records or testimony. You generally should not give an adverse insurer a recorded statement before obtaining legal advice.
Your own policy may require reasonable cooperation, so do not ignore your insurer or make a false statement. The duties owed to your own carrier differ from the lack of duty owed to an opposing carrier. Counsel can review the policy, prepare you for any required statement, attend when appropriate, or handle communications on your behalf.